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Consider two variables x and y that do not commute. Monomials xiyjxkyl can be represented by paths on a lattice in R2 starting at the origin. Similarly for n variables. Therefore, a polynomial is represented by a summation over paths γ.

We want to consider the continuous version of this. Imagine subdividing the lattice; allow non-integer powers xi1 m and let m go to infinity. Now let xi1 m=e1 mzi in the complex algebra of power series A.

A is a ‘connection’. For Ω=∑zidti let Eγ(z) be the holonomy of Ω. Now we can define the NC Fourier Transform using this NC exponential.

Kapranov went on to consider the problem for higher dimensional membranes instead of paths. For a lattice box in the variables x1 and x2 there is now a 2-cell filling the box
x1 x2 ⇒x2 x1 Introduce these xij of degree -1 in a dg-algebra B2 with d(xij)=xixj−xjxi

Let C2 be the 2-category generated by the 2-skeleton of the cubical lattice in Rn. The pasting of the half cube for C2 gives a rule in B2 . But there are 2 choices of half cube. The difference is used to extend to B3 by adding xijk now of degree -2 with d(xijk)=[xij,xk]+[xj,xik]+[xjk,xi]

so B3 is the universal enveloping algebra of a dg-Lie algebra. And so on … onto the continuous version of this.

Phew. Must be off.

Marni

P.S. Browser options here aren’t great. We apologise for errors that we are unable to correct at this stage.

Back again. More on Kapranov…

For the full dg-algebra B we have generators xI of degree (−p+1 ) where I is an index set of p elements. The differential
d(xI)=∑I=J∐Jε(J,K)[xJ,xK] (where ε is the sign of the shuffle) satisfies d2 =0 .

Theorem: B is a free NC resolution of C[x1 ,⋯,xn] and with respect to d the cohomology is
Hj(B)=C[x1 ,⋯,xn]
for j=0 and zero otherwise.

Apparently the proof uses that the Lie algebra is a Harrison complex of a free graded commutative algebra, but don’t ask me what that is.

We want to realise C2 inside B, but the horizontal pasting of two diamonds gives two possible results, leading to a definition of
D2 as the quotient of B by d of the commutators [xij,xkl]. Then C2 quotiented by translations fits into D2 .

Sigh. Now the continuous version. Extend the NC power series in z1 ⋯zn by zij, zijk and so on, as above. Let
Ω=∑zidt+∑i<jzijdtidtj+⋯ of total degree 1. The claim is that dΩ+1 2 [Ω,Ω]=0 .

What has this got to do with connections on gerbes? Let G0 acting on G−1 be a 2-group (Urs has mentioned these often enough). For gi the Lie algebras we have a dg-Lie algebra in degrees -1 and 0. Now let M be a manifold and F=dΩ+1 2 [Ω,Ω]=F2 +F3 for F2 ∈ΩM2 ⊗g0 and similarly for F3 .

If F=0 then it so happens that for all membranes σ there exists an H(σ) in the universal enveloping algebra of g with dH(σ)=H(γ1 )−H(γ2 ) which apparently only depends on σ up to reparameterization.

Then onto Chen’s holonomy, but I’m going to skip that bit. To cut a long story short: smooth membranes in Rn correspond to the universal enveloping algebra of the appropriate Lie algebra quotiented by the commutator piece like above.

What else happened yesterday? Well - they opened the university bar for us - but seriously: Berger talked about Iterated Wreath Products with theorems such as a Quillen equivalence between TopastΔop and just Topast. He then described a recursive family for n-fold loop spaces: start with the one point set dense in Set. Apply the magic wreath business and get something dense in Cat, and so on. Then simplicial objects in the nth iterate happen to form a classifying topos for n-discs with the unique arrow being ‘convex subcells of trees’. For those who know more about this than me, these Quillen equivalences have the left adjoint a n-fold Segal functor.

We also had some relatively physical talks. Bondal spoke about derived categories of toric varieties.

Posted at July 12, 2005 11:27 PM UTC

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Read the post Kapranov and Getzler on Higher StuffWeblog: The String Coffee TableExcerpt: Lecture notes of talks by Kapranov on noncommutative Fourier transformation and by Getzler on Lie theory of L_oo algebras.Tracked: June 22, 2006 8:04 PM